On the glassy nature of random directed polymers in two dimensions

@article{Mzard1990OnTG,
  title={On the glassy nature of random directed polymers in two dimensions},
  author={Marc M{\'e}zard},
  journal={Journal De Physique},
  year={1990},
  volume={51},
  pages={1831-1846}
}
  • M. Mézard
  • Published 1 September 1990
  • Physics
  • Journal De Physique
We study numerically directed polymers in a random potential in 1 + 1 dimensions. We introduce two copies of the polymer, coupled through a thermodynamic local interaction. We show that the system is unstable versus an arbitrary weak repulsion of the two copies. This suggests a similarity with a spin glass phase, with several « valleys », where the typical differences of the free energies of the valleys grow like t03C9, where t is the length of the polymer and 03C9 is probably equal to 1/3. The… 

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References

SHOWING 1-10 OF 16 REFERENCES
Directed polymers in a random medium: 1/d expansion
The problem of direct polymers in a random medium has a mean-field theory similar to, but simpler in form than, spin glasses. We propose here a method which allows one to expand the free energy of
Polymers on disordered trees, spin glasses, and traveling waves
We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions
Thermal fluctuations in some random field models
Exact identities are derived for a family of models including (a) a domain wall in a random field Ising model (RFIM), and (b) the random anisotropyXY model in the no-vortex approximation. In
Scaling of directed polymers in random media.
  • Kardar, Zhang
  • Materials Science
    Physical review letters
  • 1987
TLDR
Directed polymers subject to quenched external impurities (as in a polyelectrolyte in a gel matrix) are examined analytically, and numerically to suggest a superuniversal exponent of $\ensuremath{\nu}=\frac{2}{3}$.
Dynamic scaling of growing interfaces.
TLDR
A model is proposed for the evolution of the profile of a growing interface that exhibits nontrivial relaxation patterns, and the exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations.
Diffusion of directed polymers in a random environment
We consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time. In spatial dimensionsd>2, we establish that the behavior is
Order parameter for spin-glasses
An order parameter for spin-glasses is defined in a clear physical way: It is a function on the interval 0-1 and it is related to the probability distribution of the overlap of the magnetization in
On a mechanism for explicit replica symmetry breaking
We show that explicit replica symmetry breaking can be implemented by averaging over a small random magnetic field the partition function to an appropriate power. These results give a new insight on
...
...